The approximate message passing (AMP) algorithm [1]}, [2]} has been proposed for low-complexity implementation of SBL [3]}, [4]}. AMP was originally developed for compressive sensing based on loopy belief propagation (BP) [2]}. Compared to convex optimization based algorithms such as LASSO [6]} and greedy algorithms such as iterative hard-thresholding [7]}, AMP has low complexity and its performance can be rigorously characterized by a scalar state evolution (SE) in the case of a large independent and identically distributed (i.i.d.) (sub-)Gaussian matrix \(\mathbf { A }\)  [8]}. AMP was later extended in [9]} to solve general estimation problems with a generalized linear observation model [10]}. By implementing the E-step using AMP in the expectation maximization (EM) blackbased SBL method, matrix inversion can be avoided, leading to a significant reduction in computational complexity. However, AMP does not work well for a generic matrix such as non-zero mean, rank-deficient, correlated, or ill-conditioned matrix \(\mathbf {A}\)  [11]}, resulting in divergence and poor performance.
